The 21-year-old got the opener after just three minutes, but Todd Cantwell equalised almost immediately.
Mason Mount restored the lead after 17 minutes, but the in-form Teemu Pukki continued his good run just past the half-hour mark.
The Blues dominated the second half, and Abraham's second was enough for the win.
Chelsea had earned just a single point from their first two matches and lost the UEFA Super Cup final to Liverpool on penalties on August 14.
Chelsea had to deal with some adversity before the match even started, as Pedro injured himself warming up and was replaced by Ross Barkley in the starting XI.
The match got off to a flying start, with two goals inside the first six minutes. First, Abraham scored his first-ever goal for the Chelsea senior side, blasting home a volley after a good ball in from Cesar Azpilicueta.
But the visiting fans had barely stopped celebrating when the score was tied. The in-form Pukki teed up Cantwell, who beat goalkeeper Kepa Arrizabalaga.
Pukki has been vital for the Canaries early in the season:
The match continued at a fast pace, and before 20 minutes had passed, the visitors struck again. Mount, a standout during the early stages, took advantage of some sloppy play at the back to score his second in two matches.
Christian Pulisic got the assist, per sports writer Nizaar Kinsella:
Andreas Christensen nearly doubled the lead with a diving header, but goalkeeper Tim Krul made a fine stop to deny him.
The Blues gradually took control of proceedings but once again failed to keep out Norwich after 30 minutes, and this time, it was Pukki himself who got the goal.
His start to life in the Premier League has been sensational:
The setback handed all momentum to Norwich, who started to threaten for a lead of their own. Kepa had to be alert to keep out Emiliano Buendia, and the rebound from Grant Hanley also couldn't beat the goalkeeper.
The final chance of the half fell to the Blues, but Barkley's shot was a poor one and never threatened Krul.
The second half started with a poor shot from Mount, and Chelsea regained control of proceedings, with Norwich lacking fluency coming out of the dressing room. The Canaries defended well, however, blocking multiple shots at the last second.
And when the defenders were beaten, Krul was there to save the day with a great save on Emerson.
The video assistant decided against awarding a penalty after Marco Stiepermann made contact with Azpilicueta, and NewsUK's Jake Heasman was not happy about it:
But minutes later, the Blues finally found the breakthrough. The Blues showed their athleticism on the break, and Abraham notched his second of the contest.
The Blues defence continued to struggle, and Ben Godfrey nearly punished their poor play with a third goal but hit the crossbar.
Chelsea had a goal ruled out after substitute Olivier Giroud backed into Krul, forcing him to drop a cross. The decision seemed a harsh one, but VAR deemed the contact strong enough to award a foul. The system was also used after Godfrey hit Mount on the ankle, but he didn't receive a second yellow card.
Mount left the pitch with a late knock, but the setback didn't break Chelsea's focus. The team easily saw out the five minutes of added time, winning their first competitive match under Lampard.